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For a solution \( S \) of the \( n \)-queens problem, let \( M(S) \) denote the maximum of the absolute values of the diagonal numbers of \( S \), and let \( m(S) \) denote the minimum of those absolute values. For \( n \geq 4 \), let \( F(n) \) denote the minimum value of \( M(S) \), and let \( f(n) \) denote the maximum value of \( m(S) \), as \( S \) ranges over all solutions of the \( n \)-queens problem. Say that a solution \( S \) is an \( n \)-\emph{champion} if \( M(S) = F(n) \) and \( m(S) = f(n) \).
Approximately linear bounds are given for \( F(n) \) and \( f(n) \), along with computational results and several constructions together providing evidence that the bounds are excellent. It is shown that, in the range \( 4 \leq n \leq 24 \), \( n \)-champions exist except for \( n = 11, 16, 21, 22 \).